--- title: Stochastic Generating Function NN (SGFNN) url: https://www.emergentmind.com/topics/stochastic-generating-function-neural-network-sgfnn type: topic --- # Stochastic Generating Function NN (SGFNN) The stochastic generating function neural network (SGFNN) is a neural network model for learning stochastic Hamiltonian systems from observational data. SGFNN preserves symplectic structure of the underlying stochastic Hamiltonian system and produces symplectic predictions. The model uses an autoencoder framework to identify the randomness of the latent system by the encoder network, and detects the stochastic generating function of the system through the decoder network based on the random variables extracted from the encoder. Symplectic predictions are then generated by the stochastic generating function [2507.14467]. ## 1. Stochastic Hamiltonian setting SGFNN is formulated for \(2d\)-dimensional Stratonovich stochastic Hamiltonian systems of the form \[ \begin{aligned} d p &= f(p,q)\,dt + \sum_{k=1}^r \sigma_k(p,q) \circ dW_k(t), \quad p(t_0)=p_0, \\ d q &= g(p,q)\,dt + \sum_{k=1}^r \gamma_k(p,q) \circ dW_k(t), \quad q(t_0)=q_0. \end{aligned} \] Here \(p,q \in \mathbb{R}^d\), and \(W_k\), \(k=1,\dots,r\), are independent standard Brownian motions. The system is Hamiltonian in the stochastic sense if there exist smooth functions \(H(p,q)\) and \(H_k(p,q)\), \(k=1,\dots,r\), such that \[ \begin{aligned} f(p,q) &= -\frac{\partial H}{\partial q}(p,q)^T, &\quad \sigma_k(p,q) &= -\frac{\partial H_k}{\partial q}(p,q)^T, \\ g(p,q) &= \phantom{-}\frac{\partial H}{\partial p}(p,q)^T, &\quad \gamma_k(p,q) &= \phantom{-}\frac{\partial H_k}{\partial p}(p,q)^T. \end{aligned} \] This formulation covers additive and multiplicative noise, separable and non-separable Hamiltonians, and systems with single or multiple noises [2507.14467]. The solution flow \(\varphi_t(x_0;\omega)\), with \(x_t=(p_t^T,q_t^T)^T\), is symplectic almost surely. If \(D\varphi_t := \frac{\partial \varphi_t}{\partial x_0}\), then \[ (D\varphi_t)^T J D\varphi_t = J, \quad \text{a.s.}, \qquad J=\begin{bmatrix} 0 & I \\ -I & 0 \end{bmatrix}. \] A learned or numerical one-step map \(\widetilde{\varphi}: x_n \mapsto x_{n+1}\) is symplectic if it satisfies the same condition. In the SGFNN construction, this property is imposed by design rather than recovered only empirically [2507.14467]. The examples used to instantiate the framework include a linear stochastic oscillator with additive noise, the Kubo oscillator with multiplicative noise, a non-separable stochastic Hamiltonian system with \[ H_0 = \tfrac12 (p^2+1)(q^2+1), \qquad H_1 = 0.1(p+q)^2, \] and synchrotron oscillations with two multiplicative noises. These examples are chosen to span distinct structural regimes of stochastic Hamiltonian dynamics [2507.14467]. ## 2. Stochastic generating functions and random symplectic maps In symplectic geometry, a generating function is a scalar function whose partial derivatives define a symplectic map. For the first kind, a function \(S(P,q)\) defines the map \((p,q)\mapsto (P,Q)\) through \[ p = P + \frac{\partial S}{\partial q}^T, \qquad Q = q + \frac{\partial S}{\partial P}^T. \] In the stochastic setting, the generating function becomes a random function \(S(P,q,\omega)\). If \(\varphi_\omega:(p,q)\mapsto(P,Q)\) is a smooth random map, then \(\varphi_\omega\) is symplectic if there exists locally a smooth random map \(S(P,q,\omega)\) such that \(\frac{\partial (P^T q + S)}{\partial P \partial q}\) is invertible almost surely and \[ p = P + \frac{\partial S}{\partial q}^T(P,q,\omega), \qquad Q = q + \frac{\partial S}{\partial P}^T(P,q,\omega), \qquad \text{a.s.} \] This is the structural device used by SGFNN to ensure that the learned one-step map is symplectic [2507.14467]. For a time step \(\Delta\), the exact one-step solution map of an SHS is a random symplectic map \(\varphi_\Delta\). SGFNN represents it locally by a stochastic generating function \[ S_\Delta(P,q;\text{noise}) \equiv S(P,q,z), \] where \(z\sim\mathcal N(0,I)\) is a finite-dimensional latent representation of the stochastic forcing over \([t,t+\Delta]\). The paper does not derive an explicit analytic formula for \(S\); instead, \(S(P,q,z)\) is treated as an unknown function to be learned [2507.14467]. A common terminological confusion is to read “generating function” here as a probabilistic generating function. In SGFNN, the term refers to the symplectic-geometry generating function of a random map. This suggests that the model is fundamentally a structure-preserving discretization of stochastic Hamiltonian flow rather than a generic stochastic latent-variable network. ## 3. Autoencoder architecture and learning objective For a fixed time step \(\Delta\), the training data are one-step pairs \[ \mathcal{D} = \{(x_0^{(i)},x_1^{(i)})\}_{i=1}^M, \qquad x_j^{(i)}=(p_j^{(i)T},q_j^{(i)T})^T \in \mathbb{R}^{2d}. \] SGFNN assumes that for each pair there exists a latent Gaussian variable \(z^{(i)}\) such that the pair is generated by an implicit symplectic map induced by a stochastic generating function [2507.14467]. The encoder is \[ z^{(i)} = E_\Delta(x_0^{(i)},x_1^{(i)};\theta_1), \qquad z^{(i)}\in\mathbb R^{n_z}. \] The latent variable is trained to satisfy three conditions: it should be independent of \(x_0\), distributed as standard Gaussian, and have dimension \(n_z\) matching the number of effective stochastic drivers. For a batch \(B=\{z^{(i)}:i=1,\dots,K\}\), the distribution-matching loss is \[ \mathcal{L}_D(B)=\mathcal{L}_{\text{distance}}(B)+\tau\,\mathcal{L}_{\text{moment}}(B), \] with \[ \mathcal{L}_{\text{moment}}(B)= \sum_{j=1}^6 \frac{1}{c_j}\left\|\widehat{\mu}^{(j)}(B)-\mu^{(j)}(P_{\text{normal}})\right\|_2^2 + \frac{\nu}{V}\sum_{1\le j